Theorem imaeq2 4942

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4879	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4833	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4617	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4617	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2224	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1343   ran crn 4605   |` cres 4606   "cima 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  imaeq2i  4944  imaeq2d  4946  ssimaex  5547  ssimaexg  5548  isoselem  5788  f1opw2  6044  fopwdom  6802  ssenen  6817  fiintim  6894  fidcenumlemrk  6919  fidcenumlemr  6920  sbthlem2  6923  isbth  6932  ennnfonelemp1  12339  ennnfonelemnn0  12355  ctinfomlemom  12360  ctinfom  12361  tgcn  12848  iscnp4  12858  cnpnei  12859  cnima  12860  cnconst2  12873  cnrest2  12876  cnptoprest  12879  txcnp  12911  txcnmpt  12913  metcnp3  13151